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Exponential and Logarithmic Functions

Concepts: • Rules of Exponents • Exponential Functions – Power Functions vs. Exponential Functions – The Definition of an Exponential Function – Graphing Exponential Functions – Exponential Growth and Exponential Decay • Compound Interest • Logarithms – Logarithms with Base a ∗ ∗ ∗ ∗ ∗

Definition Exponential Notation vs. Logarithmic Notation Evaluating Logarithms Graphs of Logarithms Domain of Logarithms

– Properties of Logarithms ∗ Simplifying Logarithmic Expressions ∗ Using the Change of Base Formula to Find Approximate Values of Logarithms • Solving Exponential and Logarithmic Equations (Chapter 5)

10.1

Rules of Exponents

The following are to remind you of the rules of exponents. You are expected to know how to use them. To review, see section 5.1 in your textbook. Let c be a nonnegative real number, and let r and s be any rational numbers. Then • cr cs = cr+s

• (cr )s = crs

cr = cr−s , (c 6= 0) cs √ • c1/s = s c

•

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1 , (c 6= 0) c√r √ r s r = c = ( s c)

• c−r = • cr/s

10.2

Exponential Functions

Example 10.1 (Power Functions vs. Exponential Functions) • Sketch the graphs of y = P (x) = x2 and y = E(x) = 2x on the same graph. • Sketch the graphs of y = P (x) = x3 and y = E(x) = 3x on the same graph.

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In the previous example, both of the P functions are power functions, and both of the E functions are exponential functions. • What are the characteristics of a power function?

• What are the characteristics of an exponential function?

We can see that power functions are very different than exponential functions, so we should expect to treat them in very different ways. Solving a power equation is very different than solving an exponential equation. Finding the inverse function (if there is one) of a power function is very different than finding an inverse function of an exponential function. Do not be confused because both types of functions have exponents. It matters if the variable is in the base or in the exponent. 2

10.2.1

The Graphs of Exponential Functions

Example 10.2 (Exponential Graphs) The graphs of y = f (x) = 3x , y = g(x) = 1.5x , y = h(x) = 0.5x and y = k(x) = 0.2x are drawn below for you. Label the graphs with their function names. Compare and contrast the graphs. 6 5 4 3 2 1 0 -5

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What are some characteristics of the graph of y = f (x) = ax if a > 1?

What are some characteristics of the graph of y = f (x) = ax if 0 < a < 1?

What happens if you try to graph y = b(x) = (−2)x ?

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What happens if you try to graph y = c(x) = 1x ?

What happens if you try to graph y = d(x) = 0x ?

10.2.2

Understanding Exponential Functions

Definition 10.3 (Exponential Functions) Let a be a positive number that is not equal to one. The exponential function with base a is a function that is equivalent to f (x) = ax . NOTE: Your textbook does not tell you that a 6= 1. However, because this function behaves so differently when a = 1, most textbooks do not call g(x) = 1x an exponential function. In this course, we will follow the convention that g(x) = 1x is NOT an exponential function. Notice that b(x), c(x), and d(x) in Example 10.2 are not exponential functions. Example 10.4 (Understanding Exponential Growth) Suppose that you place a bacterium in a jar. Each bacterium divides into 2 bacteria every hour. • How many bacteria are in the jar after 2 hours? • How many bacteria are in the jar after 4 hours? • How many bacteria are in the jar after 9 hours? • Write a function to express the total number, P , of bacteria are in the jar after t hours?

• At the 30 hour mark, the jar was completely full. How many bacteria were in the jar at this time? • When was the jar half full? 4

Lots of quantities grow by a certain multiple. For example, a bacteria population model may claim that the bacteria population is doubling every hour (as in the previous example). These situations yield what is known as exponential growth models. Definition 10.5 (Exponential Growth) If a quantity, P , can be modeled by a function of the form P (t) = P0 at , where a > 1 and t represents time, then P is said to grow exponentially. Notice that P0 is the initial amount of the quantity because

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Example 10.6 A bacteria culture starts out with 1000 bacteria and doubles every 3 hours. How many bacteria will there be after 5 hours?

There is a similar phenomenon called exponential decay. This occurs when 0 < a < 1. Definition 10.7 (Exponential Decay) If a quantity, Q, can be modeled by a function of the form Q(t) = Q0 at where 0 < a < 1 and t represents time, then Q is said to decay exponentially. Exponential growth and exponential decay are, for all practical purposes, the same idea. Example 10.8 (Radioactive Decay) The half life of Actinium-225 (Ac-225) is 10 days. How much of a 30-gram sample of Ac-225 is left after one year?

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Example 10.9 JB Outlet Store is having a sale on tents. Every day a tent does not sell, its price is marked down 20%. If the price of a tent is $100 on Sunday, what is the price of the tent on Friday if it has not sold?

10.3

Compound Interest

In Definition 10.5 notice that a is the factor by which the quantity changes when t increases by one unit. This growth could also be described as a certain percentage rate which is compounded. For example, a population model may claim that the population of the earth is increasing by p% per year or an investment may grow by p% per year. When the quantity is increasing by the rate r (as a decimal) for every unit of time, then a = 1 + r, so that the model can be written as P (t) = P0 (1 + r)t . A special case of exponential growth that you are sure to run into is compound interest. When the interest is compounded yearly, there is no ambiguity. However, many times interest is compounded semiannually, quarterly, monthly, or even weekly. The interest rate is still given as an annual interest and time is usually given in years. In these cases, we must introduce another variable n which is the number of times per year the interest is compounded. Then r is the annual interest rate and t is the number of years. Then we get Proposition 10.10 (Compound Interest) If a principal P0 is invested at an interest rate r for a period of t years, then the amount P (t) of the investment is given by:  nt r P (t) = P0 1 + n

(if compounded n times per year)

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Example 10.11 (Understanding Exponential Growth) Suppose you invest $10,000 in an account that earns 5% interest compounded semiannually. • Write a function that expresses the amount of money in the account after t years. • How much money will you have in 8 years?

Example 10.12 (Understanding Exponential Growth) Suppose you invest $12,000 in an account that earns 3% interest compounded quarterly. How much money will you have in 1 year?

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In mathematics, there are a few very special numbers. The numbers 1 because they are the multiplicative and additive identities, the number π it is an irrational number that is indispensable when you discuss circles, irrational number for a multitude of reasons that you will only begin Calculus.

and 0 are special is special because and e is a special to understand in

It is important to note that the number e is a number like π is a number. It is not a variable. It is an irrational number. You never can have an exact value for e. The best you can hope to have is a decimal approximation. e ≈ 2.71828182845.... At this stage in your mathematical career, we can use compound interest to begin to explain the value of e. Example 10.13 (The number e) Suppose that you invest $1 at an annual interest rate of 100% compounded annually. How much money will you have after 1 year?

Suppose that you invest $1 at an annual interest rate of 100% compounded monthly. How much money will you have after 1 year?

Suppose that you invest $1 at an annual interest rate of 100% compounded daily. How much money will you have after 1 year?

Suppose that you invest $1 at an annual interest rate of 100% compounded every minute. How much money will you have after 1 year?

What does the value 1 +


1 n n

seem to be approaching as n becomes large?

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Suppose that you invest $1 at an annual interest rate of 100% that could be compounded continuously. How much money should you expect to have after 1 year?


n In Calculus, you will see that as n becomes very large, the quantity 1 + n1 approaches the value e. This fact can be used to justify the following formula for continuously compounded interest. Proposition 10.14 (Continuous Compounding) If P0 dollars is invested at an annual interest rate r (as decimal), compounded continuously, then the value of the investment after t years is given by P (t) = P0 ert . Example 10.15 (Continuously Compounded Interest) Jake invests $1000 at an annual interest rate of 4.6% compounded continuously. How much money will Jake have in 15 years?

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10.4

Logarithms

Exponential functions are one-to-one functions. Consequently, each exponential function has an inverse function. Why might you want to undo exponentiation? Suppose you want to solve the following equation. 10x = 3 What is happening to x? How do we undo this? Taking the xth root is not a reasonable solution. This would lead to: √ x 10 = 3 1

10 = 3 x This is even worse than before. We now have x1 as an exponent. What we need is something to pull x out of the exponent place and put it on the ground, in a manner of speaking. Logarithms are the answer. One Calculus teacher was fond of saying that logarithms are “exponent pickers.” Recall that the name of a function does not need to be a single letter. We have used f and g to mean lots of different functions. But some functions occur so regularly that it makes more sense to give them permanent names that are a bit more descriptive. This is the case with logarithms. Logarithms will have names like log, log2 , log3 , and ln. Because these are functions, they have inputs and these inputs are placed in parentheses next to the function name. For example log(x), is the output of the function named log when x is the input of the function. 10.4.1

Logarithms with base a

The logarithm with base a is the inverse function of f (x) = ax . The name of the logarithm with base a function is loga (said “log base a”). Recall that x and y trade places in inverse functions. This leads to the following definition for the logarithm with base a function. Definition 10.16 (Logarithms with Base a) Let x and y be real numbers with x > 0. Let a be a positive real number that is not equal to 1. Then loga (x) = y if and only if ay = x. In other words, the loga (x) picks the exponent to which a must be raised to produce x. There are a few special logarithms with special names. The logarithm with base 10 is most often called the common logarithm is written log(x). The logarithm with base e is most often called the natural logarithm and is written ln(x). 10

Example 10.17 (Exponential Notation and Logarithmic Notation) Convert the exponential statement to a logarithmic statement. 53 = 125

10−3 =

1 1, 000

e2 ≈ 7.389 Example 10.18 Convert the logarithmic statement to an exponential statement. log3 (35 ) = 5

log(10, 000) = 4

ln(1) = 0

Example 10.19 Evaluate each of the following. • log(100) • log(109 ) • log



1 1000



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√ • log( 3 100) • log2 (16) • ln(e5 ) • ln



• log3

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• log 1 (16) 2

Example 10.20 (Logarithm with base 2 Graph) • The graph of y = f (x) = 2x is drawn below. Sketch the graph of y = g(x) = log2 (x) on the same coordinate system. • What is the domain of g(x) = log2 (x)? What is the range of g(x) = log2 (x)? 6

y = f (x)

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What are some characteristics of the graph of y = f (x) = loga (x) if a > 1?

What are some characteristics of the graph of y = f (x) = loga (x) if 0 < a < 1?

Notice that the input of a logarithm must be greater than 0. This is one more function where you must be careful when finding the domain. Example 10.21 (Logarithm Domain) Find the domain of f (x) = log7 (2 − 5x).

Example 10.22 (Logarithm Domain) Find the domain of f (x) = ln(x2 − 4x + 3).

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10.4.2

Properties of Logarithms

Each property of logarithms is derived from the definition of the logarithm and/or a property of exponents. Property 10.23 • loga (1) = • loga (a) = • loga (ax ) = • aloga (x) = Notice that all properties can be stated in terms of the loga (x) function since ln(x) = loge (x) and log(x) = log10 (x)

Example 10.24 • Simplify ex ln(2) .

• Rewrite 5x as e to a power.

Example 10.25 Evaluate log(105 ∗ 103 ).

Property 10.26 (Product Law for Logarithms) For all u > 0 and v > 0 • loga (uv) = loga (u) + loga (v)

Proof:

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Property 10.27 (Quotient Law for Logarithms) For all u > 0 and v > 0 u = loga (u) − loga (v) • loga v Example 10.28  xy  Use the properties of logarithms to express ln as a sum and or difference of three z logarithms.

Example 10.29 Use the properties of logarithms to write the expression using the fewest number of logarithms possible. log(x2 + 2) + log(x) − log(y) − log(z)

Property 10.30 (Power Law for Logarithms) For all u > 0 and all k • loga (uk ) = k loga (u) Proof:

Example 10.31 Use the properties of logarithms to express log5 log5 (z).

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x3 √ y z



in terms of log5 (x), log5 (y), and

Example 10.32 Use the properties of logarithms to write the expression using the fewest number of logarithms possible. ln(x2 ) − 2 ln(y) − 3 ln(z)

Property 10.33 (Change of Base) If a, b, x > 0 and neither a nor b equals 1, then loga (x) =

logb (x) . logb (a)

Example 10.34 (Change of Base) Use your calculator to approximate log5 (67).

10.5

Solving Exponential and Logarithmic Equations

Remember, ”undoing” an operation means applying the inverse operation to both sides of an equation. The exponential function ax and the logarithmic function loga (x) are inverses of each other. Example 10.35 Solve. log(x + 5) = 3

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Example 10.36 Solve. log8 (x − 5) + log8 (x + 2) = 1

The properties of logarithms only work when the input is postive. If you use them to solve an equation involving logarithms, you must check your answer(s). Example 10.37 Solve. ex+2 = 5

Example 10.38 Solve.

2x − 7 = −1 3

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Example 10.39 Solve. 2x−5 = 32−2x

Example 10.40 Joni invests $1000 at an interest rate of 5% compounded monthly. When will the value of Joni’s investment reach $2500?

Example 10.41 A bacteria culture triples every 4 hours. How long until the culture doubles?

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10.6

Exponential and Logarithmic Functions Practice Problems

1. The graph of an exponential function g(x) = ax is shown below. Find a. y 6

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3. Layney invests $1000 at an interest rate of 5% per year compounded semi-annually. How much money will be in her account after 18 years? 4. Layney invests $1000 at an interest rate of 5% per year compounded continuously. How much money will be in her account after 18 years? 5. Find the average rate of change of f (x) = 2x from x = 3 to x = 4. Draw a graph that illustrates the meaning of your answer. 6. Find the exact value of the following logarithms. Do NOT use your calculator.   √ √ 1 5 3 (a) log3 (27) (b) log( 100) (c) log5 (d) ln( e3 ) (e) e2 ln(x) 25

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7. Write each expression in terms of log(x), log(y), and log(z) if possible. If it is not possible, explain why.  3 7 xy (a) log √ z  2  x + y2 (b) log z
√ (c) log x5 3 yz 8. Convert each exponential statement to an equivalent logarithmic statement. (a) 2x = 16 (b) 34 = y 9. Convert each logarithmic statement to an equivalent exponential statement. (a) log2 (32) = 5 (b) ln(x) = 3 (c) log(9) = x 10. Use your calculator to find approximate values for the following. (a) ln(7) (b) log(53) (c) log5 (6) (d) log2 (21) 11. Find all real solutions or state that there are none. Your answers should be exact. (a) 4ex−8 = 2 (b) 4x+1 = 16 (c) log8 (x − 6) + log8 (x + 1) = 1

(d) 2 log(2x) = 4

12. When a living organism dies, its carbon-14 decays. The half life of carbon-14 is 5730 years. If the skeleton of a human is discovered and has 20% of its original carbon-14 remaining, how long ago did the human die? 13. Joni invests $5000 at an interest rate of 5% per year compounded continuously. How much time will it take for the value of the investment to quadruple? 14. Joni invests $5000 at an interest rate of 5% per year compounded monthly. How much time will it take for the value of the investment to quadruple? 20

15. Let f (x) = ln(3x + 7). Find f −1 (x). 16. Let f (x) = 2x+3 − 1. Find f −1 (x). 17. Find the domain of f (x) = ln(2 − 3x) 18. Find the domain of g(x) =

x ln(5x + 4)

19. Find the domain of h(x) = ln(x2 − 2x − 15) 20. The Paper-Folding Problem Take a piece of paper and fold it in half. Now fold it in half again. And again. How many times can you fold it in half? Notice how quickly the thickness of the folded paper increases. (a) Suppose you want to write a function that models the thickness t of the paper after you fold it n times. What additional information do you need to write this function. (b) Assume that the paper you are folding is 8 21 inches by 11 inches and the paper is 0.1 mm thick. i. Write a function that models the thickness t of the paper after you fold it n times. ii. What are the units for t? iii. What are the units for n? iv. What is the domain of t(n)? (c) How thick is the folded paper after 5 folds? (d) How thick is the folded paper after 10 folds? (e) How many folds are needed for the paper to be at least as thick as a notebook? (The thickness of a notebook is about 12.8 mm.) (f) How many folds are needed for the paper to be at least as thick as an average person is tall? (The height of an average person is 1.6m.) (g) How many folds are needed for the paper to be at least as thick as the Sears tower is tall? (The height of the Sears Tower is about 440m.) (h) How many folds are needed for the paper to reach to the Sun? (The sun is about 95 million miles away. There are about 1.609344 km in 1 mile.)

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